Multiple choice

In a series of $2\mathrm{n}$ observations, half of them equal $a$ and remaining half equal $-a$. If the standard deviation of the observations is $2$, then $|\mathrm{a}|$ equals:

  1. $\displaystyle \frac{1}{\mathrm{n}}$
  2. $\sqrt{2}$
  3. $2$
  4. $\displaystyle \frac{\sqrt{2}}{\mathrm{n}}$
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C Correct answer
Explanation

Observations: n of 'a' and n of '-a'. Mean = 0. Variance = [n(a^2) + n(-a)^2] / 2n = 2na^2 / 2n = a^2. Standard deviation = sqrt(variance) = |a|. Given SD = 2, so |a| = 2.

AI explanation

For the 2n observations, the mean is calculated as (n multiplied by a plus n multiplied by -a) divided by 2n, which equals 0. The variance is the average of the squared deviations, calculated as (n multiplied by a squared plus n multiplied by a squared) divided by 2n, resulting in a squared. Since the standard deviation is given as 2, the variance is 2 squared, which equals 4. Setting a squared equal to 4 means the absolute value of a is 2.

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